1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations 15

Theorem 1 (The maximum principle). Let tlie conditions

(22) A;> 0, B; > 0, D; = Ci - Ai - Bi > 0

be fulfilled for all i = 1, 2, ... , N - 1 and the grid function Yi, Yi :f:. canst,

satisfy at all the inner nodes i = 1, 2, ... , N - 1 the condilion [,[Yi] > 0

(or [, [yi] < 0). Then Yi cannot take the maximal positive value (or the

minimal negative value) at the inner points, that is, for i = 1, 2, ... , N -1.

Proof Given a grid function Yi, let[, [y;] > 0 for all i = 1, 2, ... , N - 1

and Yi attain its maximum at one of the inner nodes i = i., 0 < i, < N,

so that

Yi * = O<i<N rnax Yi = M 0 > 0.

As Yi :f:. canst, there exists an inner point i 0 (rnay be, coincident with i.)

such that y; 0 = Yi. = M 0 > 0 and at one of the adjacent points, say for

i = i 0 - 1, we have Yio-1 < M 0. We may attempt[, [Yi] in simplified form

[,[Yi]= Bi (Yi+r - Yi) -Ai (Yi - Yi-r) - (Ci -A; - Bi) Yi,

leaving us under conditions (22) sat the point i = i 0 with

£[ Yi 0 ] = B; 0 (Yi 0 +1 - Yi 0 ) - Ai 0 (Yi 0 - Yio-d - ( Ci 0 - A; 0 - B; 0 ) Yi 0

<-Bio (Yi 0 - Yi 0 +1) -Ai 0 (Yi 0 - Yi 0 -1) < 0

by virtue of the relations

The result obtained is not consistent with the condition: [, [y;] > 0, valid
for all i = 1, 2, ... , N - 1 including i = i 0 • Thus, we proved the first
statement of the theorem. The second one can be established in a sarne
way with further replacement of Yi by -Yi.

Corollary 1 Under conditions (22) and

[, [Yi] < 0 ' i = l, 2, ... , N - 1; Yo > 0, YN > 0,

the function Yi is nonnegative: Yi > 0 for all i = 0, 1, ... , N. The case
[,[Yi] > 0, Yo < 0 and yN < 0 leads to Yi < 0 for other subscripts i =
1,2,. .. ,N-l.
Indeed, let [,[Yi] < 0 and Yi < 0 at least at one point i = i,. Then Yi
should attain its minimal negative value at an inner point i = i 0 , 0 < i 0 <
N. But this fact contradicts Theorem 1.